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Elements of Effective Math Instruction


Math learning happens best in a robust environment that attends to many needs of the learner.  It is more than just learning the concepts, practicing and testing.

The foundation for teaching is the connection we have with each student.  Teaching is an extremely personal and is predicated on relationships.  Until we establish a connection and build trust, students may not be open to learning.  This point has been driven home to me as I now have the opportunity to experience many classrooms.  Students are sometimes completely closed to the notion of learning, attempting, and taking risks.  Somehow we have to get them to trust us, to understand that we have high expectations for them, we believe in them, they can trust us to help them achieve.  We need to explicitly teach students what learning looks like; that learning in mathematics requires taking risks and making mistakes. If no one made any mistakes no one would learn anything. Mistakes and misconceptions are welcome in math class!  We can focus our dialogue and interactions around learning and improving, helping students realize that learning looks different for everyone in the room, and that in our class, everyone will grow.

Anyone who has played or coached sports knows the importance of having a clear goal in mind for any practice or training.  Every lesson in the classroom needs a clear focus that is communicated to students.  Knowing exactly what learning needs to take place keeps the teacher on target and focussed, and also is a guide to students.  The purpose (goal, target, outcome) of the lesson is the instrument against which we measure all activity in the classroom.  We use it to evaluate what lesson activities (videos, opening activities, discussions) are useful, helps focus group work (teaching students explicitly what group learning is for, holding students accountable for meaningful learning, and having students continually self-monitor their collaborative behaviour).  The goal of the lesson comes from meaningful planning, knowing our curriculum, deciding what is important, knowing what success looks like and communicating that to students.

We need to help students take ownership of the learning by constructing meaning for themselves through dialogue, rich tasks, writing and multimedia. This means interacting with the content in different ways. For example, learning to graph linear equations is concept then procedure, but can be taught through exploration using graphing software or graphing calculators, having students generate the relationships between slopes and intercepts based on their observations.  Later we ask them to communicate their learning to each other and to us through explanations and writing, rather than just performing the procedure.  There is abundant research about the importance of having students explain their reasoning rather than just performing the task.

Using multiple representations deepens mathematical understanding. Opportunities to communicate reasoning help students to understand which representations are efficient and meaningful in a given setting.  We are asking our students not only to share strategies but to then understand how to employ the most efficient strategies, both for communicating mathematically and calculating, solving, and proving.

Graphic organizers are proven to be an effective way of helping students construct meaning.  There are many types: Mind maps, frayer models, concept attainment activities, identifying similarities and differences, creating posters and diagrams.  They are effective at the beginning of a unit or lesson as review strategies (activating prior knowledge), during learning (for identifying relationships or structuring learning) or as summaries (unit review, making connections between topics).  These are just a few ideas!

The three “legs of the stool” of a firm foundation in mathematics are:  Conceptual learning, procedural fluency, and problem solving.  Each of these three is equally important! We still need to have practice and establish automaticity and efficiency.  Practice without conceptual understanding is meaningless if we are trying to create mathematically literate students that can apply and transfer their understanding.  Too often we neglect what has always been good learning in the past: direct instruction (which still has a significant effect size, especially for higher level procedural topics), and independent practice.  But all things in moderation.  A lesson can begin with whole group direct instruction (more in higher grades) and then we can also allow opportunities for collaborative learning and small group instruction, and then use activities such as graphic organizers to revisit the learning and retrieve and construct understanding.

Research from neuroscience supports the importance of writing about learning and using spaced practice as ways to solidify learning.  Formative assessment, which helps us plan and respond to learning needs and also helps students understand what they know and what they still need to know, can also help provide opportunities for spaced practice.  Everything we teach is assessed.  The assessment activities are part of the learning, not isolated events that take time away from teaching and learning.  This is why we sometimes say “assessment as learning”.

Finally, goal setting and self-monitoring help students achieve self-efficacy, which is a major predictor of success in mathematics.  Having students set learning goals for themselves is very motivating, and our journaling and formative assessment practices help students gauge their progress toward those goals .  The only way for students to set goals and measure their progress is to involve them in assessment.  We need to make assessment criteria very visible to them, even involve them in establishing the criteria.  It must be crystal clear what success in our class will look like.  A student can hit any target that they can see and that doesn’t move (Anne Davies).  This means that in every outcome there will be some dialogue and constructing assessment criteria, and having that criteria posted in the room or copied in student notebooks.

There are so many other things to consider: scaffolding, differentiating, enriching, summarizing and notetaking, explicitly teaching study skills, high level questioning, strategies for struggling math learners, vocabulary strategies, to name a few.  Please consider following me on twitter and/or google plus where I continually post research, tips, and resources to address all these areas (I rarely duplicate twitter posts on google plus, its usually one or the other).  There is so much available now to teachers and so much expected, the task seems (well let’s face it, the task is) overwhelming.  But honour what you have always done, evaluate your instruction, attend to expanding your repertoire and collection of resources, explore new methods and ideas, and above all reflect!  I wish you a year of growth and new horizons!


Negative Zero

As part of  a “connected educators” twitter/blog exploration I am part of, I am asked to answer one of two questions in a blog post.  I can write about an open ended activity that I use, or I can write about something I did that made my classroom unique, made it my own.  For the last couple of days  I was thinking about what I might write.  I love open ended stuff and have many good examples to draw from.  I was all prepared to embark on a rich description of my favourite collaborative graphing activity, when something happened.  What happened? Kenny N happened.

So running quick errands between school visits, I pull up to a stop sign, and beside me a red truck pulls up, stereo blazing, girlfriend (or wife?) in passenger seat, back of truck full of furniture.  I look over, and the shaved head, biker-looking driver is rolling down his window. The face looks familiar…I know I taught this guy…. but teaching 120 kids a semester for 11 years, well…sometimes I can’t remember their names.

Me: “Hey! How you doing?”  Him:  “Awesome, how bout you?”  Me:  “Groovy. What’s with all the furniture? You movin’? Or opening a furniture store?” Him: “Yah, and we needed a new bed, and…” Blah blah conversation continues. Both smiling, teasing, laughing.  Light turns green, we are driving side by side and yelling back and forth.  I hated to ask, but I needed to know:  Me: “Hey, remind me of your name again..?”  Him: “It’s me!  Negative Zero!  Kenny N!”  (In my defense the guy had lost a ton of weight and looked WAY different)..but it all came flooding back.

Negative zero.  I taught this kid three semesters in a row. Math was not his thing–well, school and conformity wasn’t his thing. He was a brat, in lots of trouble, but had a big heart and was just funny.

I used to post quotes by famous mathematicians around my room in an effort to immerse my students in a culture of mathematics. “We seek for truth,not only through reason, but also by the heart” -Pascal. And we’d discuss this. Was Pascal speaking as a philosopher? Or mathematician?  When we were doing geometric proofs:  “God eternally geometrizes”.  “Let no one ignorant in geometry enter here” (Plato) and “Proof is the idol before which the mathematician tortures himself”.  Others:  “This was done very elegantly by Minkowski, but chalk is cheaper than grey matter and we shall take it as it comes” Einstein, discussing what he viewed as his own weakness in mathematics. We would discuss these quotes. What was the author saying? What do they say about this crazy, historical, fanatically pursued science we call mathematics?

But the fun part was when my students would say something profound, I would write up their quote and without saying anything, put it up among the others, and wait for someone to notice. And so it was a few days after a particularly taxing debate about a problem and some interestingly misguided solutions proposed by Mr N, I put up the quote:  “The answer is Negative Zero”. -Kenny N.  He was so struck to see his words done up in fancy font and printed.  So many many years after his grad, we both still remember the debate and his crazy theory of negative zero.  And, as I said then, who knows? Maybe he’s proposing something someone will build on some day (If you are some theoretical calculus nut please don’t comment on this post some lecture on the virtues of this in terms of limits!)

Other student quotes that made the list:  After a lively debate about whether the vertices on a kite are congruent, a riled up farm kid, remembering some great hands-on activity by some teacher in middle years (thank you whoever you are!) finally shushes his arguing group by holding up a kite cut out of paper and saying “You can’t fold them corners on this one boys!”

And a mathematically gifted student who used the graphing calculators to construct stars with 6,7,8,9,10 and even 11 vertices, gathered a little cult-like following of kids who wanted to abandon graphing quadratics and just make these great stars like him (which really took a lot of thinking and many equations!).  Overheard between him and a protege: “Even- pointed stars are for the weak”.

At the time I was just being goofy, but the kids would come back to my room after they were done my class, just to see if their quote was still up. And there was much talk and excitement if someone made the “wall”.  I always encouraged them by telling them stories of mathematical discoveries made by unexpected people, sometimes young people. Like the little kid naming the Googol, the sixth graders who created a triangle much like Pascal’s, the high school drop out who proved a University professor wrong about tesselating hexagons, and so on.  I told them that Einstein felt he wasn’t a mathematician (!!Yeah, I know)  and that he said once the mathematicians got hold of his theorems, he didn’t even understand them anymore.  If we thought kids discovered something novel, we would look it up, submitted it to Universities or whatever (we actually thought we came close to a discovery with a sequence once. As it turns out we didn’t discover a new sequence but I believe we did find a novel application. I’ll tell you about it some time. )  Anyway, I pointed out that great discoveries are made by people that asked interesting questions, that Socrates said “Wonder is the beginning of  Wisdom”. I made them feel they could contribute to mathematics, that it was a vibrant, living field that was still developing, and that they could be part of it. And they remembered that.

So here’s to Kenny N, Mr. Negative Zero.  Thanks for reminding me of this!


What’s “New Math”??? Saskatchewan Curriculum

Since we renewed the math curriculum, there has been tons of press about “The New Math”, whatever that is.  Parents are frustrated, professors and other critics are saying our kids lack skills.  The Ministry even launched a review     last year inviting input from parents and teachers.  There was a parent group that formed a “WISE” chapter, (Western Initiative for Strengthening Education in mathematics), in an effort to “take matters into their own hands”.   Now we are hearing that Manitoba “abandoned” the new curriculum in favour of going back to the “basics”.  Is this  a step ahead? Who ever said we would disregard the basics anway?

Our curriculum is based on the WNCP (Western Northern Canadian Protocol), which is based on NCTM standards.  The important point here is that while NCTM advocates for research-based math instruction, which involves a more constructivist approach, no where does it say that we don’t practice basic math facts.  True, it does warn against memorization of alogorithms and procedures without context, and no one could argue that teaching math as a set of memorized steps without understanding why is good practice.  But once we have taught the concepts, given students the tools (manipulatives, diagrams, explanations) and engaged them in constructing that meaning for themselves (through dialogue, writing, explaining, peer teaching), then of course we practice.  A document that was widely distributed during curriculum change is Reflections on Research in School Mathematics: Building Capacity in Teaching a Learning, by Florence Glanfield (Published by Pearson).   Glanfield This guide contains a section on “Procedural Fluency”, citing the National Research Council’s advocacy for ensuring rigor and fluency:

Procedural fluency includes knowing the steps and rules for calculating and computing, knowing when and how to use them, and performing them with accuracy, efficiency, and flexibility. Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or soling mathematical problems“.

Glanfield quotes Sfard (2003):  “The possibility of learning mathematics without some mastery of basic procedures may be compared to the claim of building a brick house without bricks”.    While advocating for flexible strategies, NCTM also of course recognizes that students need opportunities to practice skills and procedures.  Perhaps more emphasis on teaching deeply and for understanding is “new”, but here is how NCTM advocates for that emphasis: “One of the most robust findings of research is that conceptual understanding is an important component of proficiency, along with factual knowledge and procedural facility” (Bransford, Brown, and Cocking, 1999, as cited in Principles and Satandards for School Mathematics, NCTM, 2000).  Maybe I’m missing something, but I don’t read “baby with the bathwater” here.  Our Saskatchewan Curriculum document says that math students should be “learning skills” and that “basic skills are related to Curricular Goals”.  It advocates for “appropriate” use of technology, and NCTM states that “Basic facts, processes, and translations in and among common and decimal fractions, percents, proportions, and integers are important to students’ understanding of computation”,  and does not advocate the use of a calculator until students are performing operations on numbers with several digits,  and that “students should possess adequate mental arithmetic skills so that they are not dependent on calculators to do simple computations and are able to detect unreasonable answers when using calculators” (Curriculum and Evaluation Standards, NCTM, p.96).  The renewed curriculum warns against meaningless drill, algorithms without understanding of concept or application, but it does not say we don’t practice.

Performing procedures and alogorithms is easy, not only to teach, but also to assess.  Maybe this is why procedure and computation dominated instruction in the past. But this kind of math instruction does not produce lasting learning, does not create global citizens, critical thinkers, and does not foster an appreciation for mathematics as historically rich, necessary for understanding the world around us, or useful in a life context. The average adult who studied mathematics under the older, more traditional system, was successful at math if they happened to have an innate mathematical ability or were good at memorizing.  Many of these adults even now cannot explain why certain things work the way they do. For instance, you have a greater probability of finding an eighth grader who could explain why eight divided by one half is 16 than any adult over 30.  Many more adults found math very difficult or inaccessible.

So where is the disconnect? Why has the media (and subsequently parents) decided that we don’t teach students basic facts anymore?  Perhaps the time constraints we work under are partially to blame. Teaching through inquiry and for concept attainment takes time, especially in our differentiated classrooms.  Is there time for adequate practice?  Or have we misheard the message, focusing so much on constructing meaning through multiple representations, dialogue, and manipulatives, that we have not infused enough practice? Have we felt its wrong to provide practice?  NCTM says that teachers have “pedagogical expertise”, and use professional judgement to deliver appropriate instruction. Perhaps curriculum writers didn’t emphasize skill building much because traditionally, teachers have been good at that. That’s the easy part (which is perhaps why in the past it dominated math instruction).

The answer is we need both.  Yes, we need to teach for deeper understanding, but not at the expense of providing students automaticity in basic facts and computation. If we do, they truly are at a deficit as they move up through the grades.  Students that lack procedural fluency cannot learn more complex mathematical concepts, because all their cognitive functioning is tied up with basic computations and is unavailable to grasp and construct new understandings.  Conversely, our instruction must not be dominated by “drill”, and must not ever introduce procedures without first revealing concepts and allowing students opportunity to connect and construct meaning. We need to exercise our professional discretion, knowing we have “pedagogical expertise”.

We also need to speak up for our profession and our subject. Mathematics is a wide ranging discipline, involving not only numeracy but spacial reasoning, statistical understanding, probability, logic, patterns, predictions, puzzles, technology and creative thought (to name a few!). Obviously if we only focus on computation we are not creating mathematically literate people.  Computational fluency alone will not equip our students for life in the 21st century.  We need to defend our practices by communicating that we are professionals and have valuable understanding about what is quality instruction.  All educational initiatives must be implemented through the filter of professional expertise in the context of the realities of our students, classrooms and schools,  including assessment, RTI, DI, and instruction.

Some personal thoughts:  Practice does not equal “worksheets”. We must not cave in to time saving through mindnumbing. It is practices like these that fed some people’s hatred of mathematics. Like all things, some is good, too much is not.  Especially avoid timing things (like “mad minutes”). Though some students enjoy the competition, for others this is fuel to a burning fire of mathematics anxiety. We have at our disposal now many creative ways to practice, like using small group instruction, technology, games and collaborative learning.  Call your math coach or digital learning coach for help with Destination math, ipad and computer apps.  Use these with caution also:  They can help, but effect size for web based learning as an instructional strategy, in general, is not high. “All things in moderation”.

Helping students develop mathematical fluency can be very empowering to them.  They can feel more successful and find that the speed with which they can complete math tasks will improve. Sometimes struggling math learners are helped the most by developing basic skills.  Without basic skills it is difficult for them to move ahead.

Our curriculum warns against the hazards of using rote practice as homework.  The fear is that we will send so much practice it will become “drill and kill”, and give students a negative attitude to math.  That being said, if students are open to working on developing basic skills at home, then a reasonable amount of practice would be very useful. Often this is a “comfortable” kind of home assignment, that help parents feel like they can contribute.  However:  Use with caution! A reasonable amount.  Let the parent and student have input into how much is too much!

Also, I’m going to say the dreaded “m” word: Memorization.  Truthfully, there will always be some memorization in math…but ONLY after we’ve established understanding.  Spaced practice is best, bringing concepts up again and again, reviewing, revisiting, connecting. Formative assessments can be very beneficial here, and are proven to be an effective instructional strategy.

There is really not much “new” in the “New Math”.  We are advocating for richer teaching, concept development, and authentic learning, that’s all.  We still want the kids that work at Canadian Tire after school to be able to figure out our change!  We still want to produce mathematical competence.  One word: Balance.




A new year of “PAALS” at GSSD

“Do the best you can until you know better. Then when you know better, do better”. -Maya Angelou

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Good Spirit’s PAALs process, in terms of teacher professional development, involves teachers choosing two goals and working on one of them.  Effective PD is very individual, it comes from within the individual, involves action research and experimenting, requires reflection and revision. The most powerful model of professional development involves teacher choice, and ongoing (continuous) development (as opposed to “one-hit” conference topics).  The goal is growth, not perfection! PAALS is meant to be a supportive opportunity.  Einstein said “A person who never made a mistake never tried anything new.” We have the freedom to choose an area of study and development that suits our own interests and needs.  Our PD goals are not a “deficit” model, but instead recognizes the wealth of skills and expertise we have in our teaching staff.

Professional Development goals should benefit the classroom teacher but also align with school and division goals. The role of the coach is to learn along side teachers, to help locate resources, share learnings, and have critical reflective conversations.  The process is chosen and driven by the teacher.  A coach is a collaborative colleague in the process, someone to help with whatever activities the teacher chooses to pursue, and to have reflective conversations.  A coach is never an evaluator. The target throughout is personal, professional growth, for both teacher and coach.

We are working toward a collaborative culture in our division.  We have a responsibility to share our reflective thinking process with other educational professionals, such as coaches, administrators, colleagues.

I look forward to traveling with you on your  professional growth, journey!

Success-One-Step-at-a-TimePAALS: Positive,  Accountable, Active Learning


My Commitment to My Students

Every year we start anew with a fresh group of students, knowing that these first few days are the most important, and expectations we establish (we used to call them “Class Rules”!) will set the tone for the rest of the semester.  We clarify boundaries for students, and either directly or collaboratively develop expectations for behaviour, work ethic, timelines, responsibilities and consequences.   We clearly tell our students what we expect from them, but what do they expect from us?

Teaching is a calling more than a career and it requires entering in to a very deep and deliberate relationship with people.  They put their trust in us to guide them, lead them, respect them, develop their strengths, and imbue them with dignity. Their parents trust us to not only deliver content, but to care for their children, to connect with them and foster their development.  What an honour to be trusted with these duties.  Do we communicate our intent to our students?


In the same way that we post our “class norms” or “expectations”, we can communicate our commitment to the people in our care by posting or sharing our own promises to them.  We can be transparent about the fact that we are striving to be better teachers, to serve them better. We can model that learning is lifelong and that all people have goals by sharing our professional goals in terms of how we engage with our students.  If we want to be respected, we have to deal respectfully with our students. If we want people to work toward behaviour goals, we need to model that.  If we want students to understand that they don’t have to be perfect, just strive for improvement, then let’s let them know that we strive for improvement too.

Keep a copy of your list on a neatly done poster near your teaching wall, where you can see it and be reminded, and where it is visible to students.  As you tell students what your expectations are, refer to this poster and let them know you have expectations of yourself as well, in terms of your commitment to be the best you can be for them.

Here’s an example

My Commitment to my students

  1.  Be present in the moment.  Be with each student for the full 60 minutes.
  2. Start on time. Prepare an opening each day
  3. Relate math to real life
  4. Question slowly and fairly. Use pair talk to engage girls, first nations students, and kids who don’t feel comfortable speaking up in full group. Get kids to write on whiteboard. Use peer teaching
  5. Prepare groups by reminding students what groups are for
  6. Focus on learning, enrichment, growth
  7. Reread class expectations monthly
  8. Call home about good stuff
  9. Learn about each student
  10. Document!
  11. Differentiate
  12. Call on everybody. Wait long enough. Let kids investigate, debate. Don’t deliver an answer prematurely
  13. Emphasize writing skills. Have kids verbalize processes. Share strategies
  14. Include reflection on exams
  15. Learning is about connections, not content. Reach out.

Have a good semester!!


Everyone has different aspects of instruction to work on, so our goals will be very personal. When you make your list, be thoughtful of teaching behaviours that really foster relationships with students. It’s a letter to yourself, some encouragement, and a reminder.

Be gentle with yourself.  Have a wonderful school year.


Care to share suggested items for this list? Perhaps your thoughts will inspire others. Post a reply!

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Mental Math and Estimation   [ME]

The Saskatchewan Curriculum describes this as calculating mentally and reasoning about the approximate size of quantities without calculators or pencil and paper.  It is not only estimation skill, but also computational fluency that develops efficiency and accuracy.  NCTM further describes the need for students to develop procedural fluency.  It is essential to success in mathematics.

The renewed SaskatchewanCurriculum is clear about the need to teach for deeper understanding.   Students are to be given the opportunity to understand the mathematics that underlie procedures.  We provide students opportunities to construct meaning for themselves, explore relationships through inquiry, and to represent and verbalize their understanding.  Though we may fear that taking time to allow students to create meaning around math concepts comes at the expense of developing procedural fluency, this is not the case; rather, the two are intertwined. So as long as we are providing opportunities for students to discover relationships and explore the meaning behind the math, we can confidently provide practice and expect students to develop mental recall for facts and procedures. Rather than taking away from concept exploration and deeper learning, procedural fluency enhances learning of new concepts because procedures become routine and automatic, allowing the student to focus on mathematical relationships and developing new skills.   Developing personal strategies is encouraged, but sharing and reflecting is important to help students select strategies that are efficient and accurate.

The amount of practice required to develop procedural fluency seems to be a subject of much debate.  This is a matter left to our professional discretion, understanding that procedure without context is meaningless, and the amount of practice may not be the same for every student.  Our job is to find the balance; not so much practice that it becomes meaningless and contributes to a negative perception of mathematics, but certainly enough practice to allow students to process quickly so their thinking can be focussed on new learning and understanding.


Mental Math and Estimation

Allows Students to:


  • Quickly and efficiently recall basic facts
  • Develop confidence in their math ability
  • Judge if an answer is reasonable
  • Become proficient problem solvers
  • Apply math in everyday situations

Teachers Need to:

  • Provide daily practice of math skills and estimation skills.  A few minutes of practice every day can make a difference!
  • Introduce strategies and provide practice
  • Help students understand the math behind strategies
  • Model a variety of strategies

-Nova Scotia Dept of Education


“First, enactive mastery, defined as repeated performance accomplishments (Bandura, 1982), has been shown to enhance self-efficacy more than the other kinds of cues (Bandura, 1977a, 1982; Bandura, Adams, & Beyer, 1977). Mastery is facilitated when gradual accomplishments build the skills, coping abilities, and exposure needed for task performance.”


Mental math practice develops mathematical literacy and proficiency, and prepares students for participation in a technological society.


Gist, M. E. (1987). Self-efficacy: Implications for organizational behavior and human resource

management. Academy of management review, 12(3), 472-485.

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